^ Even though I get what you mean, I still highly doubt that this was a type of transaction that would actually require a receipt of all things. From what she says, I don't think it's so much that it was legal, as it was that there were basically no laws applied. In which case a receipt would still be unnecessary, since there would be no need for legal proof of the transaction. But I guess the author may have been intending to show otherwise.
Though, even if we accept that receipts were a thing, why was that still lying arond the house? If the parents were hiding the fact that they'd bought her, shouldn't they have gotten rid of it?
Anyway, I think the real problem with this story isn't so much the believability of what is going on, as it is the way it's written. The author has clearly put a lot of thought to their characters and the world they live in, but they have a hard time sharing it with the reader, I feel. There's too much narration for a manga in my opinion (yes, I'm the type of person that doesn't like manga with too much text —when I read manga, I do it for the drawings, otherwise I'd read a novel). The concept itself is intriguing, but it is often hard to follow the story and the dialogue, I had to read a couple of parts twice to get who was speaking...